In this paper, we study data-driven localized wave solutions and parameter discovery in the massive Thirring (MT) model via the deep learning in the framework of physics-informed neural networks (PINNs) algorithm. Abundant data-driven solutions including soliton of bright/dark type, breather and rogue wave are simulated accurately and analyzed contrastively with relative and absolute errors. For higher-order localized wave solutions, we employ the extended PINNs (XPINNs) with domain decomposition to capture the complete pictures of dynamic behaviors such as soliton collisions, breather oscillations and rogue-wave superposition. In particular, we modify the interface line in domain decomposition of XPINNs into a small interface zone and introduce the pseudo initial, residual and gradient conditions as interface conditions linked adjacently with individual neural networks. Then this modified approach is applied successfully to various solutions ranging from bright-bright soliton, dark-dark soliton, dark-antidark soliton, general breather, Kuznetsov-Ma breather and second-order rogue wave. Experimental results show that this improved version of XPINNs reduce the complexity of computation with faster convergence rate and keep the quality of learned solutions with smoother stitching performance as well. For the inverse problems, the unknown coefficient parameters of linear and nonlinear terms in the MT model are identified accurately with and without noise by using the classical PINNs algorithm.

翻译：本文在物理信息神经网络（PINNs）算法的深度学习框架下，研究了大质量Thirring（MT）模型中的数据驱动局域波解与参数发现。通过相对误差和绝对误差的对比分析，精准模拟并计算了包括亮/暗型孤子、呼吸子和怪波在内的丰富数据驱动解。针对高阶局域波解，我们采用含区域分解的扩展PINNs（XPINNs）方法，捕捉孤子碰撞、呼吸子振荡及怪波叠加等完整动力学行为图像。特别地，我们将XPINNs区域分解中的界面线改进为小型界面区域，并引入伪初始条件、残差条件和梯度条件作为与各神经网络相邻接的界面条件。该改进方法成功应用于亮-亮孤子、暗-暗孤子、暗-反暗孤子、一般呼吸子、Kuznetsov-Ma呼吸子及二阶怪波等多种解。实验结果表明，该改进版XPINNs方法以更快收敛速度降低了计算复杂度，同时以更平滑的拼接性能保持了学习解的质量。对于反问题，我们利用经典PINNs算法，在有噪声和无噪声条件下精准辨识了MT模型中线性项与非线性项的未知系数参数。